Answer
dz/du = dz/dx * dx/du = 4e^x cos y * (1/u) = 4 cos y / u
dz/dy = dz/dx * dx/dy + dz/dy = 4e^x cos y * -u sin y + 4 sin y = 4(cos y - u sin y)
Work Step by Step
a) Expressing z directly in terms of u and y:
z = 4ex ln y = 4e^(ln(u cos y)) ln y = 4u cos y ln y
By using the Chain Rule:
dz/du = dz/dx * dx/du = 4e^x cos y * (1/u) = 4 cos y / u
dz/dy = dz/dx * dx/dy + dz/dy = 4e^x cos y * -u sin y + 4 sin y = 4(cos y - u sin y)
0z>0u = 4 cos y / u
0z>0y = 4(cos y - u sin y)
b) Evaluating 0z>0u and 0z>0y at the given point (u, y):
u = 2, y = p>4, x = ln (2 cos y)
0z>0u = 4 cos y / u = 4 cos (p>4) / 2
0z>0y = 4(cos y - u sin y) = 4(cos (p>4) - 2 sin (p>4))
